and have \(\mathcal {F}_{t-1}\subseteq \mathcal {F}_{t-2/3}\subseteq \mathcal {F}_{t-1/3} \subseteq \mathcal {F}_t\) .
For consistency, \(\mathcal {F}_{-1}\)  is the trivial \(\sigma \) -algebra.
Algorithm  has a generating process as follows: given \(({x}_t, {\lambda }_t)\) , we first generate \(\xi _t\)  to estimate the gradient \({\bar{g}}_t\)  and Hessian \({\bar{H}}_t\)  and derive \(\mathcal {F}_{t-2/3}\) ; then we generate \(\zeta _t\)  to obtain inexact Newton direction and derive \(\mathcal {F}_{t-1/3}\) ; then we may generate \(\psi _t\)  to select the stepsize \({\bar{\alpha }}_t\)  and derive \(\mathcal {F}_t\) . For some stepsize selection procedures like [1]}, \({\bar{\alpha }}_t\)  is fully determined by \(\xi _t, \zeta _t\)  so that no random variable \(\psi _t\)  has to be generated. In this case, we have \(\mathcal {F}_{t-1/3} = \mathcal {F}_t\) . By our setup, it is easy to see that the random quantities in Algorithm  have the following recursion
\(\sigma ({x}_t, {\lambda }_t)\cup \sigma (B_t) & \subseteq \mathcal {F}_{t-1}, \quad \hspace{8.0pt}\sigma ({\bar{g}}_t)\cup \sigma ({\bar{H}}_t)\cup \sigma ({\tilde{\Delta }}{x}_t,{\tilde{\Delta }}{\lambda }_t)\subseteq \mathcal {F}_{t-2/3},\\\sigma ({\bar{\Delta }}{x}_t, {\bar{\Delta }}{\lambda }_t) & \subseteq \mathcal {F}_{t-1/3}, \quad \sigma ({\bar{\alpha }}_t)\cup \sigma ({x}_{t+1}, {\lambda }_{t+1})\cup \sigma (B_{t+1}) \subseteq \mathcal {F}_t.\) 
